The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem


A Cohen-Macaulay complex is said to be balanced of typea=(a 1,a 2, ...,a s ) if its vertices can be colored usings colors so that every maximal face gets exactlya i vertices of thei:th color. Forb=(b 1,b 2, ...,b s ), 0≦ba, letf b denote the number of faces havingb i vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(f b )0≦ba or equivalently theh-vectors, which can arise in this way from balanced Cohen-Macaulay complexes. As part of the proof we establish a generalization of Macaulay’s compression theorem to colored multicomplexes. Finally, a combinatorial shifting technique for multicomplexes is used to give a new simple proof of the original Macaulay theorem and another closely related result.

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First and third authors partially supported by the National Science Foundation.

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Björner, A., Stanley, R. & Frankl, P. The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem. Combinatorica 7, 23–34 (1987).

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AMS subject classification (1980)

  • 05 A 15
  • 55 U 05